In ΔFGH, FH = 8ft, FG = 13ft, and m∠F = 72°
Find the length of HG.
you should get 13
Show Work.
Please Answer.
We can use the Law of Cosines to solve for the length of HG:
c² = a² + b² - 2ab cos(C)
where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.
In this case, we want to solve for HG (which is the side opposite angle F), so we can label a = FH = 8ft, b = FG = 13ft, and C = m∠F = 72°.
c² = 8² + 13² - 2(8)(13) cos(72°)
c² = 64 + 169 - 208cos(72°)
c² ≈ 169
c ≈ 13
Therefore, the length of HG is approximately 13ft, as we expected.
To find the length of HG in triangle FGH, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab cos(C)
In this case, we are looking for the length of side HG, which is side c. Side FH is length a and side FG is length b. Angle F, which is opposite side FG, is 72°.
Let's substitute the known values into the equation:
HG^2 = FH^2 + FG^2 - 2(FH)(FG)cos(F)
HG^2 = 8^2 + 13^2 - 2(8)(13)cos(72°)
HG^2 = 64 + 169 - 208cos(72°)
Now, we can calculate the value of the cosine function using a calculator or by looking it up in a trigonometric table. In this case, cos(72°) ≈ 0.309
HG^2 ≈ 64 + 169 - 208(0.309)
HG^2 ≈ 64 + 169 - 64.112
HG^2 ≈ 168.888
Finally, we can take the square root of both sides to find the length of HG:
HG ≈ sqrt(168.888)
HG ≈ 12.997 ≈ 13 (rounded to the nearest whole number)
Therefore, the length of HG is approximately 13 feet.
To find the length of HG in triangle FGH, we can use the Law of Cosines. This states that in a triangle with sides a, b, and c and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, FH is side a with length 8ft, FG is side b with length 13ft, and angle F is C with measure 72°. We want to find side c, which is HG.
Plugging in the values, we have:
HG^2 = 8^2 + 13^2 - 2 * 8 * 13 * cos(72°)
Now we can solve for HG:
HG^2 = 64 + 169 - 208 * cos(72°)
To calculate the value of cos(72°), we need to use a calculator. The value of cos(72°) is approximately 0.309, so we have:
HG^2 = 233 - 208 * 0.309
HG^2 = 233 - 64.112
HG^2 = 168.888
Taking the square root of both sides, we get:
HG ≈ √168.888
Evaluating this, we find:
HG ≈ 12.996
Rounding to the nearest whole number, the length of HG is approximately 13ft.